Hyperbolicity as an obstruction to smoothability for one-dimensional actions Christian Bonatti, Yash Lodha, Michele Triestino To cite this version: Christian Bonatti, Yash Lodha, Michele Triestino. Hyperbolicity as an obstruction to smoothability for one-dimensional actions. 2017. hal-01542608v2 HAL Id: hal-01542608 https://hal.archives-ouvertes.fr/hal-01542608v2 Preprint submitted on 17 Jul 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution| 4.0 International License Hyperbolicity as an obstruction to smoothability for one-dimensional actions Christian Bonatti∗† Yash Lodha‡ Michele Triestino† Abstract Ghys and Sergiescu proved in the 80s that Thompson’s groups F and T admit actions by C∞ diffeomorphisms of the interval. They proved that the standard actions of these groups are topologically conjugate to a group of C∞ diffeomorphisms. Monod defined a family of groups of piecewise projective homeomorphisms, and Lodha-Moore defined finitely presentable groups of piecewise projective homeomorphisms. These groups are of particular interest because they are nonamenable and contain no free subgroup. In contrast to the result of Ghys-Sergiescu, we prove that the groups of Monod and Lodha-Moore are not topologically conjugate to a group of C1 diffeomorphisms. Furthermore, we show that the group of Lodha-Moore has no nonabelian C1 action on the interval. We also show that Monod’s groups H(A), in the case where PSL(2,A) contains a rational p 1 homothety x 7→ q x, do not admit a C action on the interval. The obstruction comes from the existence of hyperbolic fixed points for C1 actions. With slightly different techniques, we also show that some groups of piecewise affine homeomorphisms of the interval are not smoothable. 1 Contents 1 Introduction 2 2 Some definitions and notation3 3 The mechanisms4 4 Historical motivation6 4.1 Thompson’s groups F and T ...............................6 2 4.2 One open problem: The Day-von Neumann problem for Diff+(R) ...........7 4.3 A second open problem: Higher rank behaviour.....................8 5 Nonamenable groups of piecewise projective homeomorphisms8 5.1 Monod’s groups.......................................8 5.2 The Lodha-Moore example................................ 10 5.3 Further examples...................................... 12 ∗CNRS. †Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, CNRS UMR 5584, 9 av. Alain Savary, 21000 Dijon, France. ‡EPFL SB MATH EGG MA C3 584 (Batiment MA) Station 8 CH-1015 Lausanne, CH-1015, Switzerland. 1MSC2010: Primary 37C85, 57M60. Secondary 43A07, 37D40, 37E05. 1 6 C1 actions of affine and piecewise affine groups 13 6.1 C1 actions of the Baumslag-Solitar groups........................ 13 6.2 The groups Gλ ....................................... 14 1 6.3 C actions of Gλ ...................................... 15 6.4 Elementary ingredients................................... 16 6.5 A particular case: no global fixed points for Aλ ..................... 16 6.6 Equivalent properties.................................... 18 6.7 General case......................................... 19 7 C2 actions with locally non-discrete stabilisers 20 7.1 Szekeres vector field.................................... 20 7.2 An obstruction to C2 smoothability........................... 20 7.3 Thompson-Stein groups.................................. 22 1 Introduction A few examples are known of groups that admit no sufficiently smooth action on a one-dimensional manifold. Following the direction of Zimmer program, typical examples come from lattices in higher rank Lie groups [9, 15, 32], or more generally from groups with Kazhdan’s property (T ) [24, 25]. Other interesting examples appear in [1, 10, 11, 26, 28]. In this work we address the problem of the existence of smooth actions of groups of piecewise projective homeomorphisms of the real line. Our principal interest comes from the existence of groups of this kind which are negative solutions to the so-called Day-von Neumann problem, as shown by Monod and Lodha-Moore [20, 23]. On the other hand, partially motivated by his work on Kazhdan groups acting on the circle, Navas raised the problem to find obstructions for a group of piecewise linear homeomorphisms of the interval to admit smooth actions (cf. [6, 26]). With this work, we illustrate relatively elementary tools which apply to a large variety of examples of such groups. Our techniques rely on some classical facts on one-dimensional dynamics and the recent work by Bonatti, Navas, Rivas and Monteverde on actions of abelian-by-cyclic groups [6]. A classical obstruction to have C1 actions on the interval is Thurston’s Stability Theorem [31]: a group of C1 diffeomorphisms of the interval is locally indicable, namely every finitely generated subgroup has a nontrivial morphism to Z. This obstruction does not apply in our setting: the group of piecewise projective homeomorphisms of the real line is locally indicable. Therefore our results exhibit new examples of locally indicable groups that have no C1 action on the interval. As an appetizer, even before introducing the notions and definitions which are necessary for presenting our main results, we start with two results whose statement are very easy to understand, and which illustrate the spirit of the paper. Fix λ > 1 and consider: • the linear map fλ : R → R defined as x 7→ λx, (x if x ≤ 0, • the map hλ : R → R defined as hλ(x) = λx if x > 0, • the translation g : x 7→ x + 1. Let Gλ be the subgroup hfλ, g, hλi ⊂ Homeo+(R). 2 Theorem 1.1. For any λ > 1 which is rational (in formula: λ ∈ Q ∩ (1, +∞)) and any morphism 1 ρ: Gλ → Diff+([0, 1]) one has : −1 the commutator [g, hλghλ ] belongs to the kernel of ρ. In particular ρ cannot be injective. 1 1 1 The same holds for any morphism ϕ: Gλ → Diff+(S ), where S is the circle. As a corollary, looking at the relations in Gλ, one obtains the stronger conclusion that for any 1 representation ρ: Gλ → Diff+([0, 1]), the image ρ(Gλ) is a metabelian group (that is, a solvable group with abelian derived subgroup). We do not know if the same occurs for λ∈ / Q. Nevertheless, consider the natural realization 1 ρ0 : Gλ → Homeo+(S ) defined as follows: 1 1 • one considers S as being the projective space RP , ! λ 0 • ρ (f ) acts on 1 as the projective action of the matrix , 0 λ S 0 1 ! 1 1 • ρ (g) acts on 1 as the projective action of the matrix , 0 S 0 1 • ρ0(hλ) coincides with ρ0(fλ) on the half circle [0, +∞] and with the identity map on the half circle [−∞, 0]. 1 1 Theorem 1.2. With the notation as above, it does not exist any homeomorphism φ: S → S so −1 −1 −1 1 1 that φρ0(fλ)φ , φρ0(hλ)φ and φρ0(g)φ belong to Diff+(S ). 1 In other words, the natural action of Gλ on S is not smoothable, and furthermore, if λ ∈ Q, then 1 every C action of Gλ on the circle or the interval are (non-faithful) metabelian actions. For more precise statements, see Theorems 6.7 and 6.9. The paper is organized as follows. In Section2 we introduce the basic objects and fix some notation. In Section3 we roughly explain the different strategies that we develop in this work, showing which are the main applications. In Section4 we illustrate the main motivation of our work, which is the recent construction by Monod of nonamenable groups without free subgroups. In Section5 we study the C1 actions of Monod’s groups and the finitely presentable group defined by Lodha-Moore. Section6 contains the main part of this work, namely the study of C1 actions of 2 the groups Gλ introduced above. Finally, in Section7 we use different techniques that work in C regularity. 2 Some definitions and notation Definition 2.1. Let M be a manifold and Homeo(M) the group of homeomorphisms of M. A group G ⊂ Homeo(M) is Cr-smoothable (r ≥ 1) if it is conjugate in Homeo(M) to a group in Diffr(M), the group of Cr diffeomorphisms of M. Remark 2.2. Even if a certain group G ⊂ Homeo(M) is not Cr-smoothable, it is still possible that the group G, as abstract group, admits Cr actions on the manifold M. 3 Throughout this work we shall only be concerned by one-dimensional manifolds. We restrict our discussion to orientation-preserving homeomorphisms, which form a subgroup Homeo+(M) of index two in Homeo(M). We will not make much distinction between the groups Homeo+(R) and r r Homeo+([0, 1]). Notice however that the groups Diff+(R) and Diff+([0, 1]) are different and for this reason we sometimes identify the interval [0, 1] to the compactified real line [−∞, +∞]. Choosing the 1 ∼ affine chart t 7→ [t : 1], we consider R as the affine line in the projective space RP = R ∪ {∞}, which 1 1 is topologically the circle S . The group Homeo+(R) can be identified to a subgroup of Homeo+(S ), 1 ∼ 1 for instance as the stabiliser of the point ∞ of S = RP . The projective special linear group PSL(2, R) = SL(2, R)/{±id} naturally acts on the projective 1 real line RP by Möbius transformations: from now on, we shall always suppose that PSL(2, R) acts on the circle in this way. 1 Definition 2.3. A circle homeomorphism h ∈ Homeo+(RP ) is piecewise projective if there exists 1 a finite partition RP = I1 ∪ ..